# Can you solve this riddle? #2 **URL:** \ **Category:** In My Humble Opinion\ **Created:** [May 4, 2015, 2:07pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240 "2015-05-04T14:07:12Z")\ **Posts on this page:** 19\ **Page:** 10
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [May 6, 2015, 4:41pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/181 "2015-05-06T16:41:59Z")
> [@Martian Bigfoot](#): > > I still don’t get why. How does Monty’s mechanism for picking the final contestant change the initial probability of the two contestants picking the set of doors with the prize at the beginning? > > If I’m just being dense, feel free to kick me. That’s what my friends and family always do. If the other guy has a car behind his door then Monty has to open yours. You lose. So let’s put that possibility away, since we want to know what happens after Monty opens the door and you’re still in the running. So - the other guy does not have a car behind his door, and Monty opens his door 'cuz he hates him. He will never open yours if that is the case 'cuz he loves you. That can happen whether you have a car behind yours or not. So the chance that you have a car behind your door is now (after Monty opened the other guy’s door) 50%. ---
**Author:** ![Don\_t\_Panic](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/don_t_panic/32/398_2.png) [@Don\_t\_Panic](https://boards.straightdope.com/u/Don_t_Panic)\ **Post date:** [May 6, 2015, 4:57pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/182 "2015-05-06T16:57:16Z")
> [@Terr](#): > > So - the other guy does not have a car behind his door, and Monty opens his door 'cuz he hates him. He will never open yours if that is the case 'cuz he loves you. That can happen whether you have a car behind yours or not. So the chance that you have a car behind your door is now (after Monty opened the other guy’s door) 50%. But in both the original scenario and the “Monty loves you” scenario we’re already assuming that the other guy has been eliminated, and you’re the one left. If Monty is “meh, whatever” about you, you know the following: You both picked a door at the beginning. The other guy is out. His door had no prize. The car is now either behind your door or behind the third door. If Monty loves you, you know the following: You both picked a door at the beginning. The other guy is out. His door had no prize. The car is now either behind your door or behind the third door. See, I just copy-pasted that, because they’re the exact same scenario. ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [May 6, 2015, 5:00pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/183 "2015-05-06T17:00:20Z")
> [@Folly](#): > > If Monty hates me and I know it. …clap your hands. ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [May 6, 2015, 5:00pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/184 "2015-05-06T17:00:59Z")
> [@Martian Bigfoot](#): > > But in both the original scenario and the “Monty loves you” scenario we’re already assuming that the other guy has been eliminated, and you’re the one left. > > If Monty is “meh, whatever” about you, you know the following: > > You both picked a door at the beginning. The other guy is out. His door had no prize. The car is now either behind your door or behind the third door. > > If Monty loves you, you know the following: > > You both picked a door at the beginning. The other guy is out. His door had no prize. The car is now either behind your door or behind the third door. > > See, I just copy-pasted that, because they’re the exact same scenario. Correct. Except in the first case the chance of car being behind your door is 2/3. And in the second case that chance is 1/2. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [May 6, 2015, 5:41pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/185 "2015-05-06T17:41:55Z")
As I hinted at earlier in thread, the probability of winning when you stick is 1/(2-A) where A is the probability that Monty reveals your door in the case that both contestants pick the wrong door. So if Monty loves you (A=0) the probability of winning by sticking is 1/2, if Monty hates you (A=1), the probability of winning by sticking is 1 and when Monty is fair (A=1/2) the probability of winning by switching is 2/3 ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [May 6, 2015, 5:45pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/186 "2015-05-06T17:45:42Z")
All this “Monty loves you/Monty hates you” stuff is only relevant if you \*know \*Monty loves you or hates you. And if Monty hated you, he’d change the game anyway. He could call an audible any time he wanted. ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [May 6, 2015, 5:52pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/187 "2015-05-06T17:52:30Z")
If Monty hates you, and you know he hates you, and Monty knows you know that he hates you, then Monty may eliminate the other guy when the car is behind the unchosen door. You will be certain you are winning, and therefore will not switch even when given the opportunity. Monty will trumph over you by having given you the chance to switch and by tricking you into thinking you had won with certainty. This is a devilish Monty indeed. ---
**Author:** ![Folly](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/folly/32/3219_2.png) [@Folly](https://boards.straightdope.com/u/Folly)\ **Post date:** [May 6, 2015, 5:53pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/188 "2015-05-06T17:53:16Z")
> [@bup](#): > > All this “Monty loves you/Monty hates you” stuff is only relevant if you \*know \*Monty loves you or hates you. > > And if Monty hated you, he’d change the game anyway. He could call an audible any time he wanted. Untrue. The probabilities remain the same. If you don’t know, it will affect how you _act_ according to what you think the probabilities are. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [May 6, 2015, 5:53pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/189 "2015-05-06T17:53:40Z")
> [@bup](#): > > All this “Monty loves you/Monty hates you” stuff is only relevant if you \*know \*Monty loves you or hates you. > > And if Monty hated you, he’d change the game anyway. He could call an audible any time he wanted. When I worked out the answer to the problem I left A as its value isn’t fixed by the problem as stated. The point I made earlier was that when A is unknown the best strategy is still to stick. You don’t know if sticking gives you an advantage, but you do know that you won’t be disadvantaged by sticking. ---
**Author:** ![Folly](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/folly/32/3219_2.png) [@Folly](https://boards.straightdope.com/u/Folly)\ **Post date:** [May 6, 2015, 5:57pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/190 "2015-05-06T17:57:32Z")
> [@Asympotically\_fat](#): > > When I worked out the answer to the problem I left A as its value isn’t fixed by the problem as stated. > > The point I made earlier was that when A is unknown the best strategy is still to stick. You don’t know if sticking gives you an advantage, but you do know that you won’t be disadvantaged by sticking. This is true. So, I guess my added sentence from earlier is not needed. Since the question asks “Should this contestant switch?” The answer will always be no. ETA: Also, apologies for apparently skipping over your posts before. ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [May 6, 2015, 6:00pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/191 "2015-05-06T18:00:20Z")
> [@Folly](#): > > Untrue. The probabilities remain the same. If you don’t know, it will affect how you _act_ according to what you think the probabilities are. Discussing probabilities about which you have no information doesn’t do much for me. Equally as uninteresting would be, “If the car is behind the 3rd curtain, but you didn’t know it, your best strategy would be to pick curtain 3, although you wouldn’t know that.” ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [May 6, 2015, 6:51pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/192 "2015-05-06T18:51:16Z")
Let’s look at the named-goat scenario with a spiteful but dumb Monty, who will always get rid of you if he can, OR will always get rid of your opponent if he can. > [@Left\_Hand\_of\_Dorkness](#): > > _the goats have names_. Clarice and Boomer. > Here are six possibilities: > > 1. You picked the right prize, Bob picked Clarice. > 2. You picked the right prize, Bob picked Boomer. > 3. Bob picked the right prize, you picked Clarice. > 4. Bob picked the right prize, you picked Boomer. > 5. You picked Clarice, Bob picked Boomer. > 6. You picked Boomer, Bob picked Clarice. Things change in these circumstances: In scenario 1 and 2, Monty is gonna reveal Bob’s goat. Right? You’ll get a chance to switch. In scenario 3 and 4, Monty is gonna reveal your goat. Right? You’ll be out of the game. In scenario 5 and 6, Monty has to choose which goat to reveal. Here is where his hatred comes in: if he hates you, he’s always going to eliminate you, and if he loves you, he never will. Do you agree with me so far? If he hates you, the only scenarios where you stay in the game are 1 and 2: in either scenario, you MUST stay or you lose. If he hates you, you stay in the game in scenarios 1 and 2 and 5 and 6. If it’s 1 or 2, you should stay; if it’s 5 or 6, you should switch. This gives you a 50/50 chance of success. ---
**Author:** ![Folly](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/folly/32/3219_2.png) [@Folly](https://boards.straightdope.com/u/Folly)\ **Post date:** [May 6, 2015, 7:04pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/193 "2015-05-06T19:04:08Z")
> [@bup](#): > > Discussing probabilities about which you have no information doesn’t do much for me. Equally as uninteresting would be, “If the car is behind the 3rd curtain, but you didn’t know it, your best strategy would be to pick curtain 3, although you wouldn’t know that.” I don’t really understand the point you are making. You don’t “know” that Monty is choosing randomly in the OP. Unless you add my sentence from before. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [May 6, 2015, 7:10pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/194 "2015-05-06T19:10:59Z")
To understand why this different from the original Monty Hall problem: Your chance of picking the right door first time is 1/3. What Monty tells you (in this variation) by revealing your opponents door is either: i) Nothing (when Monty loves you A=0) ii) Your first pick is correct (when Monty hates you A=1) iii) Your first pick was probably correct (0\ **Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [May 6, 2015, 7:49pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/195 "2015-05-06T19:49:20Z") Just to be clearer: in this variation you win 1/3 of the time by sticking _including those times where Monty reveals your door (and hence you lose before being given the option to switch or stick)_. ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [May 6, 2015, 8:17pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/196 "2015-05-06T20:17:28Z")
> [@Terr](#): > > I will repost my earlier post, modified for “Monty hates me” twist: > > There are three possible initial scenarios: > > 1. You: car, Bob: nothing, third door: nothing - probability 1/3 > 2. You: nothing, Bob: nothing, third door: car - probability 1/3 > 3. You: nothing, Bob: car, third door: nothing - probability 1/3 > > Let’s see what the scenarios become when Monty opens the door: > > 1. You: car, Bob: nothing, third door: nothing, Monty HAS to open Bob’s door- probability still 1/3 > 2a. You: nothing, Bob: nothing, third door: car, Monty picks Bob’s door to open - probability is 0 since Monty hates you > 2b. You: nothing, Bob: nothing, third door: car, Monty picks your door - probability is 1/3 > 2. You: nothing, Bob: car, third door: nothing, Monty HAS to open your door - probability still 1/3 > > From the four above, 2a and 3 are removed, since you are eliminated in those scenarios, and the original question presupposes that you stay to the next round. So what is left is 1 and 2b. The chance of 1 happening is the same as that of 2b. So it doesn’t matter if you switch or not. I think there’s a mistake here. In 2b, you’re eliminated as well, so the only remaining option is 1. If Monty hates you, always stay. > [@Asympotically\_fat](#): > > As I hinted at earlier in thread, the probability of winning when you stick is 1/(2-A) where A is the probability that Monty reveals your door in the case that both contestants pick the wrong door. > > So if Monty loves you (A=0) the probability of winning by sticking is 1/2, if Monty hates you (A=1), the probability of winning by sticking is 1 and when Monty is fair (A=1/2) the probability of winning by switching is 2/3 I agree with the formula, but I think the very last statement was accidentally switched. Where A=1/2, the probability of winning by \*\*sticking \*\*is 2/3. Given that, it doesn’t matter whether Monty loves you or hates you. As long as he HAS to open a door picked by a contestant that doesn’t have a prize (even if he gets to pick which one if there’s a tie), you should always stick with your first door (because at best switching is neutral, at worst it’s a guaranteed loss and if things are random it lowers your odds). ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [May 6, 2015, 8:27pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/197 "2015-05-06T20:27:43Z")
> [@Quercus](#): > > I agree with the formula, but I think the very last statement was accidentally switched. Where A=1/2, the probability of winning by \*\*sticking \*\*is 2/3. > > Given that, it doesn’t matter whether Monty loves you or hates you. As long as he HAS to open a door picked by a contestant that doesn’t have a prize (even if he gets to pick which one if there’s a tie), you should always stick with your first door (because at best switching is neutral, at worst it’s a guaranteed loss and if things are random it lowers your odds). Yes that’s correct, unfortunately I have the nasty habit of altering my posts a lot before posting, but missing a few words here and there. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [May 6, 2015, 8:36pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/198 "2015-05-06T20:36:57Z")
> [@Biotop](#): > > Good job to **bup** who first answered that refusing to trade was the right answer. Sonuvagun - I thought I was first! But looking back at the first page, **bup** changed his mind while I was drafting my post. I was still the first one to give that answer in the poll, though. 🙂 ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [May 6, 2015, 9:14pm UTC](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240/199 "2015-05-06T21:14:06Z")
> [@RTFirefly](#): > > Sonuvagun - I thought I was first! But looking back at the first page, **bup** changed his mind while I was drafting my post. > > I was still the first one to give that answer in the poll, though. 🙂 And kudos to you as well, **RTFirefly**! This was especially tricky given the history of the Monty riddle here. [Previous page](https://boards.straightdope.com/t/can-you-solve-this-riddle-2/719240.md?page=9)